### III.3 Five Factors in Eight Runs

II.4 Sixteen Run Fractional
Factorial Designs






Introduction
Resolution Reviewed
Design and Analysis
Example: Five Factors Affecting
Example / Exercise: Seven Factors
Affecting a Polymerization Process
Discussion
II.4 Sixteen Run Fractional
Factorial Designs:
Introduction

With 16 runs, up to 15 Factors may be analyzed
at Resolution III.
– Resolution IV is possible with 8 or fewer factors.
– Resolution V is possible with 5 or fewer factors.
These designs are very useful for “screening”
situations: determine which factors have strong
main effects
 20% rule

II.4 Sixteen Run Designs:
Resolution Reviewed
Q: What is a Resolution III design?
confounded with other main effects, but at least
one main effect is confounded with a 2-way
interaction
 Resolution III designs are the riskiest fractional
factorial designs…but the most useful for
screening
– “damn the interactions….full speed ahead!”

II.4 Sixteen Run Designs:
Resolution Reviewed
Q: What is a Resolution IV design?
confounded with other main effects or 2-way
interactions, but either (a) at least one main
effect is confounded with a 3-way interaction, or
(b) at least one 2-way interaction is confounded
with another 2-way interaction.
 Hence, in a Resolution IV design, if 3-way and
higher interactions are negligible, all main effects
are estimable with no confounding.

II.4 Sixteen Run Designs:
Resolution Reviewed

Q: What is a Resolution V design?
with other main effects or 2- or 3-way interactions, and
2-way interactions are not confounded with other 2way interactions. There is either (a) at least one main
effect confounded with a 4-way interaction, or (b) at
least one 2-way interaction confounded with a 3-way
interaction.
II.4 Sixteen Run Designs:
Resolution Reviewed

Hence, in a Resolution V design, if 3-way and
higher interactions are negligible, all main effects
and 2-way interactions are estimable with no
confounding.
16 Run Signs Table
Actual
Order
Sum
Divisor
Effect
y
16
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
BD
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
CD
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
ABC
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
ABD
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
ACD
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
BCD
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
ABCD
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
II.4 Sixteen Run Designs
Example: Five Factors


y = clip time (secs) for 16 parts from the sprue (injector
for liquid molding process)
Factors and levels
+
– A: Table
No
Yes
– B: Shake
No
Yes
– C: Position
Sitting
Standing
– D: Cutter
Small
Large
– E: Grip
Unfold
Fold
*Contributed by Rodney Phillips (B.S. 1994), at that time
working for Whirlpool. This was a STAT 506 (Intro. To
DOE) project.
Example: Five Factors

Design the Experiment: associate factors
with carefully chosen columns in the 16run signs matrix to generate a design
matrix
– Always associate A, B, C, D with the
first four columns
– With five factors, E = ABCD is
universally recommended (or E= ABCD)
Example: Five Factors
Full Alias Structure for the design E=ABCD
I=ABCDE
A=BCDE
B=ACDE
C=ABDE
D=ABCE
E=ABCD
AB=CDE
AC=BDE
AE=BCD
BD=ACE
BE=ACD
CD=ABE
CE=ABD
DE=ABC
Example: Five Factors
Completed Operator Report Form
Std.
Order
12
16
3
2
4
13
5
6
9
1
10
11
8
7
14
15
Actual
Order
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A=
Table
Yes
Yes
No
Yes
Yes
No
No
Yes
No
No
Yes
No
Yes
No
Yes
No
B=
Shake
Yes
Yes
Yes
No
Yes
No
No
No
No
No
No
Yes
Yes
Yes
No
Yes
C=
Position
Sitting
Standing
Sitting
Sitting
Sitting
Standing
Standing
Standing
Sitting
Sitting
Sitting
Sitting
Standing
Standing
Standing
Standing
D=
Cutter
Large
Large
Small
Small
Small
Large
Small
Small
Large
Small
Large
Large
Small
Small
Large
Large
E=
Grip
Unfold
Fold
Unfold
Unfold
Fold
Fold
Unfold
Fold
Unfold
Fold
Fold
Fold
Unfold
Fold
Unfold
Unfold
y = Clip
Time (s)
46.30
27.35
54.89
40.05
28.82
45.99
57.69
29.49
44.19
31.55
28.47
29.16
36.01
39.51
36.60
52.41
Example: Five Factors
Completed Signs Table with Estimated Effects
Actual
Order
10
4
3
5
7
8
14
13
9
11
12
1
6
15
16
2
Sum
Divisor
Effect
y=
clip
time
31.55
40.05
54.89
28.82
57.69
29.49
39.51
36.01
44.19
28.47
29.16
46.30
45.99
36.60
52.41
27.35
628.5
16
39.28
A
B
C
D
AB
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-82.4
8
-10.3
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
0.40
8
0.05
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
21.6
8
2.70
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
-7.52
8
-0.94
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
7.36
8
0.92
AC
BC
BD
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
-1
1
1
-1
-1
1
1
-50.0 16.24
8
8
-6.25 2.03
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
-29.4
8
-3.68
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
-0.48
8
-0.06
CD
ABC
=DE
ABD
=CE
1
-1
-1
1
1
1
1
1
1
1
-1
-1
-1
1
-1
-1
-1
1
-1
-1
1
-1
1
-1
-1
-1
1
-1
1
-1
-1
1
-1
-1
-1
1
1
1
1
1
-1
-1
1
-1
-1
1
1
1
6.88 10.72 26.96
8
8
8
0.86 1.34 3.37
ACD
=BE
BCD
=AE
ABCD
=E
-1
-1
1
-1
-1
1
1
1
1
1
-1
1
1
-1
-1
-1
1
1
-1
1
1
-1
-1
-1
-1
-1
1
-1
-1
1
1
1
-21.8 18.08
8
8
-2.72 2.26
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-107.8
8
-13.48
Example: Five Factors
Normal Plot of Estimated Effects
Ordered
Effects:
-13.48
-10.28
-6.25
-3.68
-2.72
-0.94
-0.06
0.05
0.86
0.92
1.34
2.03
2.26
2.70
3.37
AC=BDE
A=BCDE
E=ABCD
-20
-10
Effe cts
0
10
Example: Five Factors
Preliminary Interpretation

The Normal Plot indicates three effects
distinguishable from error. These are
– E = ABCD (estimating E+ABCD)
– A = BCDE (estimating A+BCDE)
– AC = BDE (estimating AC+BDE),
marginal.
Example: Five Factors
Preliminary Interpretation
Since it is unusual for four-way
interactions to be active, the first two are
attributed to E and A
 Since A is active, the AC+BDE effect is
attributed to AC
– We should calculate an AC interaction
table and plot

Example: Five Factors
AC Interaction Table and Plot
C
1
1
2
31.55
54.89
44.19
29.16
39.95
57.69
39.51
45.99
52.41
48.90
40.05
28.82
28.47
46.30
35.91
29.49
36.01
36.60
27.35
32.26
A
2
Example: Five Factors
AC Interaction Table and Plot
Interaction Plot for y = clip time (s)
A=Table
-1 = no
1 = yes
47
42
37
32
-1=sitting
C=Position
1=standing
Example: Five Factors
Interpretation

E = -13.5. Hence, the clip time is
reduced an average of about 13.5
seconds when the worker uses the low
level of E (the folded grip, as opposed to
the unfolded grip). This seems to hold
regardless of the levels of other factors (E
does not seem to interact with anything).
Example: Five Factors
Interpretation

The effects of A (table) and C
(position) seem to interact. The
presence of a table reduces average
clip time, but the reduction is larger
(16.6 seconds) when the worker is
standing than when he/she is sitting
(4.0 seconds)
II.4 Sixteen Run Designs
Example / Exercise: Seven Factors Affecting a
Polymerization Process


y = blender motor maximum amp load
Factors and levels
– A: Mixing Speed
Lo
– B: Batch Size
Small
– C: Final temp.
Lo
– D: Intermed. Temp.
Lo
1
– F: Temp. of modifer
Lo
– G: Add. Time of modifier Lo
+
Hi
Large
Hi
Hi
2
Hi
Hi
Contributed by Solomon Bekele (Cryovac). This was part of a STAT
Example / Exercise: Seven Factors Affecting
a Polymerization Process
factors with columns of the 16-run signs
matrix


For 6, 7, or 8 factors, we assign the additional
factors to the 3-way interaction columns
For this 7-factor experiment, the following
assignment was used
E = ABC, F = BCD, G = ACD
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Runs table
Std Order
A
B
C
D
E=ABC
G=ACD
F=BCD
1
-1
-1
-1
-1
-1
-1
-1
2
1
-1
-1
-1
1
1
-1
3
-1
1
-1
-1
1
-1
1
4
1
1
-1
-1
-1
1
1
5
-1
-1
1
-1
1
1
1
6
1
-1
1
-1
-1
-1
1
7
-1
1
1
-1
-1
1
-1
8
1
1
1
-1
1
-1
-1
9
-1
-1
-1
1
-1
1
1
10
1
-1
-1
1
1
-1
1
11
-1
1
-1
1
1
1
-1
12
1
1
-1
1
-1
-1
-1
13
-1
-1
1
1
1
-1
-1
14
1
-1
1
1
-1
1
-1
15
-1
1
1
1
-1
-1
1
16
1
1
1
1
1
1
1
Example / Exercise: Seven Factors Affecting a
Polymerization Process

Determine the design’s alias structure
– There will again be 16 rows in the full alias
table, but now 27 = 128 effects (including I)!
Each row of the full table will have 8
confounded effects! Here is how to start: find
the full defining relation:
– Since E = ABC, we have I = ABCE.
– But also F = BCD, so I = BCDF
– Likewise G = ACD, so I = ACDG
– Likewise I = I x I = (ABCE)(BCDF) = ADEF !
Example / Exercise: Seven Factors Affecting
a Polymerization Process
 Continue in this fashion until you find


I = ABCE = BCDF = ACDG = ADEF = BDEG = ABFG = CEFG
We have verified that this design is of
Resolution IV (why?)
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Determine the alias table: multiply the defining relation
(rearranged alphabetically here)
I = ABCE = ABFG = ACDG = ADEF = BCDF = BDEG = CEFG
 by A for the second row:
A = BCE = BFG = CDG = DEF = ABCDF = ABDEG = ACEFG
 by B for the third row:
B = ACE = AFG = ABCDG = ABDEF = CDF = DEG = BCEFG
 and so on; after all seven main effects are done, start with
two way interactions:
AB = CE = FG = BCDG = BDEF = ACDF = ADEG = ABCEFG
and so on...(what a pain!)…until you have 16 rows.

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Full Alias Structure for the 2IV7-3 design
E = ABC, F = BCD, G = ACD
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG +
CEFG
A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG
B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG
D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF
F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG
AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG
AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF
AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG
BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Reduced Alias Structure (up to 2-way interactions)
for the 2IV7-3 design E = ABC, F = BCD, G = ACD
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG
A
B
C
D
E
F
G
(***)
AB
AC
AE
AF
AG
BD
( three-way and
+ CE +
+ BE +
+ CG +
+ BC +
+ BG +
+ BF +
+ CF +
higher
FG
DG
EF
DF
DE
CD
EG
ints.)
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Std Order
Y (amps)
A
B
C
D
E=ABC
G=ACD
F=BCD
1
130
-1
-1
-1
-1
-1
-1
-1
2
232
1
-1
-1
-1
1
1
-1
3
135
-1
1
-1
-1
1
-1
1
4
235
1
1
-1
-1
-1
1
1
5
128
-1
-1
1
-1
1
1
1
6
184
1
-1
1
-1
-1
-1
1
7
133
-1
1
1
-1
-1
1
-1
8
249
1
1
1
-1
1
-1
-1
9
130
-1
-1
-1
1
-1
1
1
10
225
1
-1
-1
1
1
-1
1
11
143
-1
1
-1
1
1
1
-1
12
270
1
1
-1
1
-1
-1
-1
13
132
-1
-1
1
1
1
-1
-1
14
198
1
-1
1
1
-1
1
-1
15
138
-1
1
1
1
-1
-1
1
16
249
1
1
1
1
1
1
1
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Completed Signs Table with Estimated Effects
Actual
Order
y=
max
amps
A
B
C
D
Unknown
130
232
135
235
128
184
133
249
130
225
143
270
132
198
138
249
2911
16
181.9
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
773
8
96.6
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
193
8
24.1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-89
8
-11.1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
59
8
7.4
Sum
Divisor
Effect
AB
AC
=CE =BE
=FG =DG
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
135
8
16.9
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
-75
8
-9.4
=CG
=EF
BC
=AE
=DF
BD
=CF
=EG
CD
=AG
=BF
ABC
=E
ABD
ACD
=G
BCD
=F
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
25
8
3.1
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
61
8
7.6
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
37
8
4.6
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
-13
8
-1.6
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
75
8
9.4
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
19
8
2.4
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-15
8
-1.9
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-63
8
-7.9
ABCD
=AF
=BG
=DE
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-49
8
-6.1
Example / Exercise: Seven Factors Affecting a
Polymerization Process

Analyze the Experiment: as an exercise,
– construct and interpret a Normal probability
plot of the estimated effects;
– if any 2-way interactions are distinguishable
from error, construct interaction tables and
plots for these;
– provide interpretations
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Solution: Normal Plot of Estimated Effects
Ordered
Effects:
-11.1
-9.4
-7.9
-6.1
-1.9
-1.6
2.4
3.1
4.6
7.4
7.6
9.4
16.9
24.1
96.6
A
B
-20
0
20
Effe cts
60
80
100
Example / Exercise: Seven Factors Affecting a
Polymerization Process
Suggested Interpretation



The effect of mixing speed is A = 96.6 amps.
Hence, when we change the mixing speed from
its low setting to its high setting, we expect the
amps.
The effect of batch size is B = 24.1 amps.
Hence, when we change the batch size from
small to large, we expect the motor’s max amp
None of the other factors seems to affect the
II.4 Discussion


As in 8-run designs, we can always “fold over” a
16 run fractional factorial design. There are
several variations on this technique; in particular,
for any 16-run Resolution III design, it is always
possible to add 16 runs in such a way that the
pooled design is Resolution IV.
There are a great many other fractional factorial
designs; in particular, the Plackett-Burman
designs have runs any multiple of four
(4,8,12,16,20, etc.) up to 100, and in n runs can
analyze (n-1) Factors at Resolution III.
II.4 References




Daniel, Cuthbert (1976). Applications of Statistics
to Industrial Experimentation. New York: John
Wiley & Sons, Inc.
Box, G.E.P. and Draper, N.R. (1987). Empirical
Model-Building and Response Surfaces. New
York: John Wiley & Sons, Inc.
Box, G.E.P., Hunter, W. G., and Hunter, J.S.
(1978). Statistics for Experimenters. New York:
John Wiley & Sons, Inc.
Lochner, R.H. and Matar, J.E. (1990). Designing
for Quality. Milwaukee: ASQC Quality Press.